OPTIMAL PLACEMENT OF ACTUATORS

IN FIBER REINFORCED POLYMER

COMPOSITE SHELL STRUCTURES

USING GENETIC ALGORITHM

A THESIS SUBMITTED IN PARTIAL REQUIREMENTS FOR THE

DEGREE OF

Bachelor of Technology

In

Mechanical Engineering

By

ANJANA SATPATHY

Roll No. – 10503075

Department of Mechanical Engineering

National Institute of Technology, Rourkela

2009

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National Institute of Technology

Rourkela

CERTIFICATE

This is to certify that this report entitled, “OPTIMAL PLACEMENT OF

ACTUATORS IN FIBER REINFORCED POLYMER COMPOSITE

SHELL STRUCTURES USING GENETIC ALGORITHM” submitted

by Anjana Satpathy in partial fulfillments for the requirements for the award

of Bachelor of Technology Degree in Mechanical Engineering at National

Institute of Technology, Rourkela (Deemed University) is an authentic work

carried out by her under my supervision and guidance.

To the best of my knowledge, the matter embodied in this

report has not been submitted to any other University / Institute for

the award of any Degree or Diploma

Date:

NIT Rourkela (Prof. T. Roy)

Dept. of Mechanical Engineering,

National Institute of Technology

Rourkela - 769008, Orissa

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ACKNOWLEDGEMENT

I deem it a privilege to have been a student of Mechanical

Engineering stream in National Institute of Technology, Rourkela

I express my deep sense of gratitude and obligation to my project

guide Prof. T. Roy for his invaluable guidance and support. I am very

grateful to him for allowing me to do this project and for his constant help

and support throughout the making of this project.

Anjana Satpathy

Roll No. – 10503075

8th

Semester, B.TECH

Department of Mechanical Engineering

National Institute of Technology, Rourkela

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ABSTRACT

Active vibration control of smart FRP composite structures finds use in high

performance structures especially in light weight composite structures.

Proper implementation of such smart structure systems demands complete

understanding of their responses, optimal placement of sensors and

actuators, and design of an appropriate control system. In the present work,

an improved genetic algorithm (GA) based optimal collocated sensors and

actuators of smart fiber reinforced polymer (FRP) composite shell structures

has been presented. Layered shell finite elements have been formulated and

the formulation has been validated for coupled electromechanical analysis of

curved smart FRP composite structures having piezoelectric sensors and

actuators patches. Modal analysis has been performed to transfer the coupled

finite element equation to state space equation. An integer-coded GA-based

open-loop procedure has been implemented for optimal placement of

actuators for maximizing controllability index. This type of GA with

uniform crossover and mutation technique has been developed to efficiently

search for optimal locations of sensors/actuators.In this project, we have

used integer coded GA to find optimal placement of actuators on spherical

shell structures and semi-circular ring.

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CONTENTS

ABSTRACT 4

1. INTRODUCTION 8

1.1 Fiber Reinforced Polymer 11

2. LITERATURE REVIEW 13

2.1 Parameter Variation 13

2.2 Deterministic Methods 14

2.3Stochastic Methods 17

2.4 Motivation and Objectives 21

3. GA FOR OPTIMAL PLACEMENT OF ACTUATORS 23

3.1 Genetic Algorithm 23

3.2 Shell Finite Element 29

3.3 State Space Representation 31

3.4 GA for Optimal Placement 34

4. RESULTS AND DISCUSSION 37

4.1 Structural validation 37

4.2 Validation for optimal actuators placement 37

4.3 Optimal vibration control of a semi-circular ring 40

4.4 Optimal vibration control of laminated spherical shell panel 42

5. CONCLUSION AND SCOPE FOR FUTURE WORK 45

5.1 Conclusion 45

5.2 Scope for Future Work 45

6. REFERENCES 46

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LIST OF FIGURES

1.Schematic representation of the basic elements of a Smart Structure 9

2. Figure representing Basic Genetic Algorithm 28

3. Smart layered shell element 29

4. Shell element with various coordinates system 30

5. Optimal location of four actuators on the beam substrate based

on maximum controllability 39

6. Comparison of variation of controllability index with generation

for the cantilever beam using integer- and binary-coded GA 40

7. Actuators location on the semicircular ring based on maximum

controllability index 41

8. Variation of controllability index with generation for semicircular

ring 41

9. Schematic Representation of a Spherical shell 43

10. Actuators location on the spherical panel based on

maximum controllability index. 43

11.Variation of controllability index with generation for spherical

panel 44

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LIST OF TABLES

1. Material properties of structural laminate and PZT 38

2. Several important parameters for integer- and binary-coded GA 39

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CHAPTER 1

INTRODUCTION

Active vibration control in distributed structures is of practical interest

because of the demanding requirement for guaranteed performance. This is

particularly important in light-weight structures as they generally have low

internal damping. An active vibration control system requires sensors,

actuators, and a controller. The design process of such a system

encompasses three main phases such as structural design, optimal placement

of sensors and actuators and controller design. In vibration suppression of

structures, locations of sensors and actuators have a major influence on the

performance of the control system. It is well known that misplaced sensors

and actuators lead to problems such as the lack of observability or

controllability. Active vibration control is defined as a technique in which

the vibration of a structure is reduced by applying counter force to the

structure that is appropriately out of phase but equal in force and amplitude

to the original vibration. As a result two opposing forces cancel each other,

and structure essentially stops vibrating. Techniques like use of springs,

pads, dampers, etc have been used previously in order to control vibrations.

These techniques are known as ‘Passive Vibration Control Techniques’.

They have limitations of versatility and can control the frequencies only

within a particular range of bandwidth. Hence there is a requirement for

‘Active Vibration Control’. ‘Active Vibration Control’ makes use of ‘Smart

Structures’. This system requires sensors, actuators, a source of power and a

compensator that performs well when vibration occurs. Smart Structures are

used in bridges, trusses, buildings, mechanical systems, space vehicles,

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telescopes, and so on. The analysis of a basic structure can help improve the

performance of the structures under poor working conditions involving

vibrations. “A Smart Structure” means a structure that can sense an external

disturbance and respond to that with active control in real time to maintain

the mission requirements. A Smart Structure typically consists of a host

structure incorporated with sensors and actuators coordinated by a

controller. The integrated structured system is called Smart Structure

because it has the ability to perform self diagnosis and adapt to

environmental change. One promising application of such smart structure is

the control and suppression of unwanted structural vibrations. Fig. 1 depicts

the schematic representation of the basic elements of a smart structure

Fig. 1 Schematic representation of the basic elements of a Smart

Structure

• Sensor Patch

It is bonded to the host structure (Beam). It is generally made up of

piezoelectric crystals (one of the smartest materials). It senses the

disturbance of the beam and generates a charge which is directly

proportional to its strain. Direct piezoelectric effect is used here.

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• Controller

The charge developed by the sensor is given to the controller. The controller

lines the charge according to suitable control gain and then the charge is fed

to the actuator. Controller also forms the feed back transfer function for this

system.

• Actuator Patch

The lined up charge from the controller is fed to the actuator. An actuator is

a piezoelectric patch bonded to the host. Due to the input voltage, actuator

causes pinching action (or generates shear force along the surface of the host

which acts as the damping force and helps in the attenuating vibration

motion of the beam. Converse piezoelectric effect is used here.

A given structure can vibrate with many modes. The design of controller for

all the modes is very difficult. However, all the modes do not contribute

significantly to the overall disturbance. Hence, we filter out the modes which

cause the maximum disturbance. Hence a controller can be designed to

control only these modes.

In this project we have used an improved integer coded GA along with

improved uniform crossover and mutation technique for determination of

optimal placement of sensors and actuators. Optimal placement of PZT

actuators on the curved smart FRP composite structures (i.e. semicircular

ring, spherical and ellipsoidal shell panel) have been studied based on the

controllability index, which is the singular value of the control input matrix.

Higher the controllability index, the lower will be the electrical potential

required for active control. Integer coded genetic algorithm has been applied

to efficiently find the maximum controllability index.

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1.1 FIBER REINFORCED POLYMER

A Fiber Reinforced Polymer (FRP) composite is defined as a polymer

(plastic) matrix, either thermoset or thermoplastic, that is reinforced

(combined) with a fiber or other reinforcing material with a sufficient

aspect ratio (length to thickness) to provide a discernable reinforcing

function in one or more directions. FRP composites are different from

traditional construction materials such as steel or aluminum. FRP

composites are anisotropic (properties apparent in the direction of the

applied load) whereas steel or aluminum is isotropic (uniform properties in

all directions, independent of applied load). Therefore, FRP composite

properties are directional, meaning that the best mechanical properties are in

the direction of the fiber placement. Composites are similar to reinforced

concrete where the rebar is embedded in an isotropic matrix called concrete.

Composition

Composites are composed of:

o Resins - The primary functions of the resin are to transfer stress

between the reinforcing fibers, act as a glue to hold the fibers

together, and protect the fibers from mechanical and environmental

damage. The most common resins used in the production of FRP

grating are polyesters (including orthophthalic-“ortho” and

isophthalic-“iso”), vinyl esters and phenolics.

o Reinforcements - The primary function of fibers or reinforcements is

to carry load along the length of the fiber to provide strength and

stiffness in one direction. Reinforcements can be oriented to provide

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tailored properties in the direction of the loads imparted on the end

product. The largest volume reinforcement is glass fiber.

o Fillers - Fillers are used to improve performance and reduce the cost

of a composite by lowering compound cost of the significantly more

expensive resin and imparting benefits as shrinkage control, surface

smoothness, and crack resistance.

o Additives - Additives and modifier ingredients expand the usefulness

of polymers, enhance their processability or extend product durability

Each of these constituent materials or ingredients play an important role in

the processing and final performance of the end product.

There are a wide variety of processes available to the composites

manufacturer to produce cost efficient products like pultrusion and various

other molding processes.

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CHAPTER 2

LITERATURE REVIEW

Vibration suppression performance in both active and passive

damping decisively depends on the number, shape, size and location of the

piezoelectric ceramic elements used as sensors and actuators [1,2]. The same

holds for shape control, vibroacoustic control and structural health

monitoring. Depending on the complexity of the structure, analytic or

numerical models might prove more appropriate to describe its behavior. A

number of different objective functions, design variables, constraints and

solution methods can be applied for the optimization of a target application.

The following subsections reviews a representative portion of the work

performed in the last decade towards the optimal placement of sensors and

actuators for vibration suppression. The articles reviewed here have been

classified based on the optimization algorithm used.

2.1 Parameter Variation

Informal optimization consisting of parameter variation studies can

deliver useful insight into the optimization task, in particular if the solution

space can be explored with a reasonable number of configurations. This is

the case for simple structures such as beams. While investigating the

multiple mode passive vibration suppression with piezoelectric materials and

resonant shunts, Hollkamp [3] estimated the generalized electromechanical

coupling coefficient of a pair of piezoelectric ceramic tiles attached to a

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cantilever beam at different locations. Kang et al [4] optimized the

placement of piezoelectric collocated sensor/actuator pairs for active

vibration control of laminated beams by maximizing the structural damping

index, a weighted sum of the achieved modal damping of each vibrational

mode. Parametric studies were presented for the damping ratio as a function

of the location of piezoelectric ceramic elements with given length and

various outer-layer fiber orientations. Vibration suppression analysis of

cantilever beam with piezoelectric sensors/actuators subjected to an exciting

force has been performed by Zhang and Kirpitchenko [5]. They considered

two sets of surface bonded piezoelectric patches with three locations of

patches and experimentally showed that the damping of combined beam-

piezoelectric patches system increased by 8-10 times in comparison to that

of mechanical system. Formal optimization techniques, on the other hand,

can be classified into deterministic methods and stochastic methods.

2.2 Deterministic Methods

Most mathematical programming methods work locally and are very

efficient given that the assumptions on continuity, differentiability and

convexity of the solution space are satisfied. Aside from the convexity

assumption, this is mostly the case for basic structures such as beams and

plates. Classical beam and plate structural models were used to derive cost

functions for determining the optimum placement and thickness of

embedded and surface mounted piezoactuators by Main et al. [6]. An

optimization procedure was used to develop a design guide for simplified

determination of piezoactuator size and placement. Li et al [7] presented an

optimal design methodology for piezoelectric ceramic actuators/sensors and

feedback gains towards the vibration suppression in flexible structures and

studied the influence of the actuator/sensor pairs on the mass and stiffness

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properties of the composite structure. The proposed composite objective

function included the control performance as well as the added mass.

However, the gradient based optimization methods applied to the simple

case of a beam structure, was prone to getting trapped in local optima. Kang

et al [8] carried out an investigation on laminated plates where the

optimization was carried out using the gradient method. Haramoto et al. [9]

presented the optimal placement of two pairs of sensors and actuators in

order to maximize the H2 norm of the closed loop system for a simply

supported beam using quasi-Newton method. Mukherjee and Joshi [10]

obtained the actuator layout by minimizing the power consumption in order

to achieve a specified displacement of plate structure using iterative

procedure. Wang and Wang [11] proposed a controllability index for optimal

locations and size of piezoelectric actuators for the beam model in order to

maximize modal control forces and reported that higher the controllability

index, the smaller would be the electrical potential required for active

control. However, they did not consider control spillover of the higher order

modes, which would give closed loop instability by maximizing modal

control forces of the higher order modes. Seeger and Gabbert [12] proposed

an optimization algorithm for the optimal positioning of collocated

actuator/sensor patch pairs on a simply supported plate structure. Conjugate

gradient method was applied to minimize the H2-norm of the transfer

function between an external excitation disturbance and the plate vibration

amplitude. The constrained optimization algorithm used the augmented

Lagrangian function in order to avoid patch overlapping. The quasi-modal

sensor and quasi-modal actuator were developed for finding optimal

placement and sizes of sensors and actuator on rectangular plate by Sun et al

[13]. Sun and Tong [14] extended the investigation to simply supported

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closed- and open-form cylindrical shell structures. An energy based

approach for optimal positioning of piezoelectric actuators and sensors on a

flexible structure was presented by Leleu et al. [15]. First, a two-dimensional

(2–D) model of a piezoelectric actuator bonded to a plate was obtained and

then, a Ritz formulation was used to find a state model of the system in view

of its control. Selection process for piezoelectric transducers (PZT) used as

actuator elements for suppressing vibrations in a flexible beam system was

discussed by Kermani et al. [16]. The effects of changing physical

parameters such as the relative thickness of the piezoelectric ceramic with

respect to the beam, the optimum location of the PZT actuator, and the

length of the PZTs were studied based on the singular value decomposition

of the controllability Grammian of the resulting system. Modal based

correction methods were applied by Rose [17] for the placement of

piezoelectric ceramic modules on a circular plate. These methods allow the

negotiation of changes introduced by the piezoelectric element’s mass and

stiffness. The generalized electromechanical coupling coefficient was

maximized by applying gradient-based methods in a two-step approach.

Halim and Moheimani [18] suggested a criterion for the optimal placement

of collocated piezoelectric ceramic actuator/sensor pairs on a thin plate using

modal and spatial controllability. The spatial controllability was used to find

the optimal placement of collocated actuator/sensor pairs for effective

average vibration reduction over the entire structure, while maintaining

modal controllability and observability of selected vibration modes. Sun and

Tong [19] presented an investigation into design optimization of actuator

patterns for static shape control of composite plates with piezoelectric

actuator patches. An energy optimization based method for finding the

optimal control voltages that can actuate a structure shape close to the

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desired one within a given error was described. Emilio et al. [20] proposed a

simultaneous search for an optimal topology of a flexible structure as well as

the optimal position of the piezoceramic in the design. The method was

implemented based on the SIMP (‘Solid Isotropic Material with

Penalization’) material model and the examples presented were limited to

two-dimensional models.

2.3 Stochastic Methods

Engineering design problems, are often of a discrete nature (e.g. the

number of actuators), so that the above methods described in the previous

subsection are not applicable or tend to get trapped in local optima. In order

to overcome these limitations, the scientific community has put significant

effort into the investigation of stochastic optimization methods. Stochastic

optimization methods can handle search spaces involving both discrete and

continuous domains, non-convex objective functions, and objective

functions or constraints lacking differentiability. A drawback is that

stochastic search methods are often computationally expensive. Genetic

algorithm (GA) has been extensively used for optimization of engineering

problems in recent times and some of the important works in this direction

are described here. Rao et al [21] were the first to apply genetic algorithms

to the problem of optimal actuators placement in an actively controlled two-

bay truss. The dissipation energy of the active controller was maximized for

a fixed number of three actuators. A strategy for determining the optimal

number of actuators and their respective locations in the active vibration

control of a 72-bar space truss was presented by Yan and Yam [22] where

the eigenvalues of the energy correlative matrix of the input control force

were used to determine an optimal number of actuators for vibration control.

They reported that depending on the desired controllability level, these can

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be equal to or less than the number of degrees of freedom to be controlled.

Using a binary-encoded genetic algorithm, Bishop and Striz [23]

demonstrated the optimal placement of passive ideal viscous dampers on

space trusses subjected to different loading. The kinetic and strain energy

remaining in a system at the end of a full time-domain transient analysis, as

well as the number of actuators, were combined to form a penalty function.

Abdullah et al. [24] used genetic algorithm to simultaneously place

collocated sensor/actuator pairs in multi-storey building while using output

feedback as the control law in terms of minimizing the quadratic

performance i.e. weighted energy of the system. They found optimal gain

using Davidon-Fletcher-Powell gradient-based optimization algorithm by

choosing weighting matrices [Q] and [R] using trial and error and concluded

that the decision variables in this optimization problem were greatly

dependent on the selection of weighting matrices. They also used binary

coded GA with the length of the gene string as the number of floors in multi-

storey building, which led to large number of function evaluations and large

number of generations to reach near optimal solution. Richardson and

Abdullah [25] used a real-encoded genetic algorithm for optimal placement

of sensors and active tendon mechanisms on high-rise civil structures which

were susceptible to vibrations due to earthquakes, hurricanes or other

abnormal loads such as explosions. The proposed method allowed for the

simultaneous determination of the optimal controller gains. However, real-

encoded genetic algorithm is more suitable for continuous search space

where structural responses are obtained analytically. Results by Gaudenzi et

al [26] provided insight into the problem of optimal placement, sizing and

loading of piezoelectric actuators for damping beam vibrations. A

fundamental solution, formulated for a single piezoelectric actuator pair, was

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used in the framework of a genetic algorithm optimization. A float-encoded

genetic algorithm for the integrated optimization of piezoelectric actuator

and sensor locations and feedback gains for active vibration control was

introduced by Zhang et al [27] and concluded that the float-encoded genetic

algorithm was less likely to become trapped in local minima compared to the

adaptive binary genetic algorithm and converged faster to the solution. A

cantilever beam was presented as an optimization example, for which the

performance function is based on maximizing the dissipation energy of the

active controller. However, float-encoded genetic algorithm was also

appropriate for continuous search space. A similar problem was tackled by

Yang et al [28] and they presented a simultaneous optimization method

considering several design variables such as placement of collocated

piezoelectric sensors/actuators, size of sensor/actuator and feedback control

gain for vibration suppression of simply supported beam by minimizing the

equivalent total mechanical energy of the system. However, they did not

consider input energy in the used objective function i.e. equivalent total

mechanical energy. This type of chromosome representation used will not be

feasible for multi input system with more sensors and actuators and it will

also lead to more trial and error to impose bound for the entire feedback

control gain matrix elements. The same authors later extended the method

cited above to the investigation of plates and cylindrical shells [29] with

dynamic constraints, included directly in the modified real-encoded genetic

algorithm, and penalizes overlapping piezoelectric patches. Binary coded

genetic algorithms based on the open loop performance were used by Han

and Lee [30] to find efficient locations for six sensors and two actuators out

of 99 possible sub-areas on a cantilever composite plate. Two criteria for the

optimal placement of piezoelectric actuators for vibration control were

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suggested by Sadri et al [31] using modal controllability and the

controllability Grammian. The number of actuators, their sizes and their

optimal locations for maximum controllability of isotropic plates were

determined using genetic algorithms. They used Gray coded genetic

algorithm to find the eight coordinates of two piezoelectric actuators in a

simply supported plate based on the open loop performance. However, this

type of Gray coded GA leads to increased string length. The authors later

applied the modal controllability as a criterion for optimal placement of

piezoelectric actuators for panel flutter suppression [32]. Quek et al [33]

used the classical direct pattern search method to maximize the active

damping of a laminated composite plate. The starting point for the pattern

search was selected based on the maxima of integrated normal strains

consistent with the size of the collocated piezoelectric sensor/actuator pair

used. Optimization performance indices were based on modal and system

controllability. Guo et al. [34] presented a sensor placement optimization

performance index based on the damage detection in the two dimensional

truss structures using binary coded genetic algorithm. Li et al. [35] proposed

two level genetic algorithms (TLGA) for optimal placement of active tendon

actuators in multi storey building by minimizing the maximum top floor

displacement. This proposed TLGA might be feasible for this type of

optimization problem and for active vibration control of large-scale

structures with complete electromechanical analysis considering PZT

sensors/actuators but this will not be computationally feasible because there

will more possible actuators locations. The positions of four piezoelectric

patches for adaptive feed-forward control were chosen out of 64 candidate

locations on a cantilever aluminum plate by [36] and concluded that the

maximization of the controllability Grammian through a genetic algorithm

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guaranteed a minimum control force for minimizing the vibration response

at three selected points of the plate. Wang et al [37] addressed the topology

optimization of collocated sensors/ actuators pairs for torsional vibration

control of a laminated composite cantilever plate using output feedback

control. They used binary coded genetic algorithm for optimization, which

was not computationally efficient for actuator/sensor location in terms of

number of function evaluations, and generations for convergence. Liu et al.

[38] used a spatial H2 norm of the closed loop transfer matrix for finding the

optimal nodal points for sensing displacement and applying actuation for the

control of a fixed-fixed plate. This method did not address a complete

coupled electromechanical analysis and used binary coded genetic algorithm

leading to very large number of generations for convergence. Optimal

placements and sizes of sensors and actuators attached to an inflated torus

were found by Jha and Inman [39] using a binary encoded genetic algorithm.

Performance indices were defined using modal controllability (minimum

energy requirement) and observability (maximum output energy for a good

signal to-noise ratio). Belloli and Ermanni [40] presented optimum

placement of piezoelectric ceramic elements for vibration suppression of

rear wing of a race car. The optimization procedure included a knowledge-

based CAD model, an FE model and an evolutionary algorithm optimization

loop controlled by the proprietary software tool DynOPS.

2.4 Motivation and Objectives

Even though many works have been reported in the broad area of

active vibration control of smart structures, there are still scopes and need

for improvement in better understanding of behavior of smart shell

structures for achieving better actuation and superior control performance of

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such structures. From the exhaustive literature review, the following

important observations have been made.

i) A large number of works are available in the form of beam and plate finite

elements for analysis of piezo-laminated smart FRP structures, not many

works are available in the form of shell finite element for such structures.

ii) Many existing literatures in optimal placement of sensors/actuators have

used GA but they require large number of generations and function

evaluations for reaching near optimal solution.

Keeping the above points in mind, the specific objectives of the present

thesis have been laid down as

i) Development of a shell finite element capable of analysis coupled

electro- mechanical responses of smart FRP shell structures

ii) Development of an improved GA based optimal placement scheme

for achieving better controllability of such structures

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CHAPTER 3

GA FOR OPTIMAL PLACEMENT OF ACTUATORS

3.1 GENETIC ALGORITHM

A genetic algorithm (GA) is a search technique used in computing to

find exact or approximate solutions to optimization and search problems.

Genetic algorithms are categorized as global search heuristics. Genetic

algorithms are a particular class of evolutionary algorithms (also known as

evolutionary computation) that use techniques inspired by evolutionary

biology such as inheritance, mutation ,selection and crossover (also called

recombination).

The current framework of GAs was first proposed by Holland[41] and his

student Jong[42], and was finally popularized by another of his students,

Goldberg[43].

Genetic algorithms are implemented in a computer simulation in

which a population of abstract representations (called chromosomes or the

genotype of the genome) of candidate solutions (called individuals,

creatures, or phenotypes) to an optimization problem evolves toward better

solutions. Traditionally, solutions are represented in binary as strings of 0s

and 1s, but other encodings are also possible. The evolution usually starts

from a population of randomly generated individuals and happens in

generations. In each generation, the fitness of every individual in the

population is evaluated, multiple individuals are stochastically selected from

the current population (based on their fitness), and modified (recombined

and possibly randomly mutated) to form a new population. The new

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population is then used in the next iteration of the algorithm. Commonly, the

algorithm terminates when either a maximum number of generations has

been produced, or a satisfactory fitness level has been reached for the

population. If the algorithm has terminated due to a maximum number of

generations, a satisfactory solution may or may not have been reached.

Genetic algorithms find application in bioinformatics, phylogenetics,

computational science, engineering, economics, chemistry, manufacturing,

mathematics, physics and other fields.

A typical genetic algorithm requires:

1. a genetic representation of the solution domain,

2. a fitness function to evaluate the solution domain.

A standard representation of the solution is as an array of bits. Arrays of

other types and structures can be used in essentially the same way. The main

property that makes these genetic representations convenient is that their

parts are easily aligned due to their fixed size, which facilitates simple

crossover operations. Variable length representations may also be used, but

crossover implementation is more complex in this case. Tree-like

representations are explored in genetic programming and graph-form

representations are explored in evolutionary programming.

The fitness function is defined over the genetic representation and

measures the quality of the represented solution. The fitness function is

always problem dependent. For instance, in the knapsack problem one wants

to maximize the total value of objects that can be put in a knapsack of some

fixed capacity. A representation of a solution might be an array of bits,

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where each bit represents a different object, and the value of the bit (0 or 1)

represents whether or not the object is in the knapsack. Not every such

representation is valid, as the size of objects may exceed the capacity of the

knapsack. The fitness of the solution is the sum of values of all objects in the

knapsack if the representation is valid, or 0 otherwise. In some problems, it

is hard or even impossible to define the fitness expression; in these cases,

interactive genetic algorithms are used.

Once we have the genetic representation and the fitness function defined,

GA proceeds to initialize a population of solutions randomly, then improve

it through repetitive application of mutation, crossover, inversion and

selection operators.

3.1.1 Initialization

Initially many individual solutions are randomly generated to form an

initial population. The population size depends on the nature of the problem,

but typically contains several hundreds or thousands of possible solutions.

Traditionally, the population is generated randomly, covering the entire

range of possible solutions (the search space). Occasionally, the solutions

may be "seeded" in areas where optimal solutions are likely to be found.

3.1.2 Selection

During each successive generation, a proportion of the existing

population is selected to breed a new generation. Individual solutions are

selected through a fitness-based process, where fitter solutions (as measured

by a fitness function) are typically more likely to be selected. Certain

selection methods rate the fitness of each solution and preferentially select

26

the best solutions. Other methods rate only a random sample of the

population, as this process may be very time-consuming.

Most functions are stochastic and designed so that a small proportion

of less fit solutions are selected. This helps keep the diversity of the

population large, preventing premature convergence on poor solutions.

Popular and well-studied selection methods include roulette wheel selection

and tournament selection.

3.1.3 Reproduction operator

The primary objective of the reproduction operator is to emphasize

good solutions and eliminate bad solutions in a population, while keeping

the population size constant. This is achieved by performing the following

tasks as

i) Identify good (usually above-average) solutions in a population.

ii) Make multiple copies of good solutions.

iii) Eliminate bad solutions from the population so that multiple copies of

good solutions can be placed in the population. There exist a number of

ways to achieve the above tasks. Some common methods are tournament

selection, proportionate selection, ranking selection and roulette wheel

selection.

3.1.4 Crossover Operator

The reproduction operator cannot create any new solutions in the

population. It only made more copies of good solutions at the expense of

not-so-good solutions. Creation of new solutions is performed in crossover

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and mutation operators. Like reproduction operator, there exists a number of

crossover operators in the GA literature, but in almost all crossover

operators, two strings are picked from the mating pool at random and some

portion of the strings are exchanged between the strings. In a single-point

crossover operator, this is performed by randomly choosing a crossing site

along the string and by exchanging all bits on the right side of the crossing

site.

3.1.5 Mutation Operator

Crossover operator is mainly responsible for the search aspect of

genetic algorithms, even though mutation operator is also used for this

purpose sparingly. The need for mutation is to keep diversity in the

population.

3.1.6 Termination

This generational process is repeated until a termination condition has

been reached. Common terminating conditions are:

• A solution is found that satisfies minimum criteria

• Fixed number of generations reached

• Allocated budget (computation time/money) reached

• The highest ranking solution's fitness is reaching or has reached a

plateau such that successive iterations no longer produce better results

• Manual inspection

• Combinations of the above

28

3.1.7 Simple generational genetic algorithm pseudo code

1. Choose initial population

2. Evaluate the fitness of each individual in the population

3. Repeat until termination: (time limit or sufficient fitness achieved)

1. Select best-ranking individuals to reproduce

2. Breed new generation through crossover and/or mutation

(genetic operations) and give birth to offspring

3. Evaluate the individual fitnesses of the offspring

4. Replace worst ranked part of population with offspring

The Flowchart below represents the Basic Genetic Algorithm

Fig. 2 Flowchart representing Basic Genetic Algorithm

29

3.2 Shell Finite Element

Layered shell finite element has been formulated for analysis of smart

laminated composite structures.In the present formulation, the kinematics

has been described using a first-order shear deformation theory based on the

Reissner–Mindlin assumptions. The basic assumptions made in the

formulation are:

(a) straight line normal to the mid-surface may not remain straight during

deformation,

(b) the strain energy corresponding to the stress component orthogonal to the

mid-surface is disregarded.

Fig. 3 shows the general smart shell element with composite and

piezoelectric layers. It has been assumed that the piezoelectric patches are

perfectly bonded to the surface of the structure and the bonding layers are

thin. The geometry and various coordinate systems of the degenerate shell

element are shown in Fig. 4.

Fig.3 Smart layered shell element

30

Fig. 4 Shell element with various coordinates system

31

3.3 State Space Representation

In control engineering, state space representation is a mathematical

model of a physical system as a set of input, output and state variables

related by first-order differential equations. To abstract from the number of

inputs, outputs and states, the variables are expressed as vectors and the

differential and algebraic equations are written in matrix form (the last one

can be done when the dynamical system is linear and time invariant). The

state space representation (also known as the "time-domain approach")

provides a convenient and compact way to model and analyze systems with

multiple inputs and outputs.

The global sets of dynamic equations for piezo- elastic analysis can be

written as

[ ] [ ] uu uu uM d K d K Fφ φ + + = && (3.1)

uK d K Gφ φφ φ + = (3.2)

The coupled piezoelectric static equations can be as follows

[ ] uu uK d K Fφ φ + = (3.3)

uK d K Gφ φφ φ + = (3.4)

For open electrodes, charge can be expressed as

0G = (3.5)

Static displacement can be calculated from the Eq. (4.6)

[ ]

uu u th

u

K K d F

GK K

φ

φ φφφ

=

(3.6)

Dynamic responses of piezolaminated structures can be calculated due to

only dynamic loading from the Eq. (3.7)

32

[ ] [ ] ..

dy dy u a aM d K d F K φ φ + = − (3.7)

where [ ]M is the overall global mass matrix, [ ]K is the overall global elastic

stiffness matrix and [ ]uaK is the global piezoelectric coupling matrices of

actuator patches. The nodal dynamic displacement vector ( )dy

d t can be

approximated by the modal superposition of the first ‘r’ modes as

( ) [ ] ( )dy

d t tψ η≈ (3.8)

where [ ]1 2[ ] ......... rψ ψ ψ ψ= is the truncated modal matrix. The decoupled

dynamic equations of Eq. (3.7) considering modal damping can be written as

( ) ( ) [ ] [ ] [ ] .. .

22 ( )

T T

i di i i i i ua at t t F Kη ξ ω η ω η ψ ψ φ

+ + = −

(3.9)

wheredi

ξ is the damping ratio.

Eq. (3.9) can be represented in state-space form as

[ ] [ ]

.ˆ

a dX A X B B uφ = + + (3.10)

[ ][ ] [ ]

[ ]2

0

2i di i

IA

ω ξ ω

=

− − is the system matrix, [ ]

[ ]

[ ] [ ]

0

T

ua

BKψ

=

− is the control

matrix, [ ]

[ ]

0ˆ

TB

Fψ

=

is the disturbance matrix, d

u is the disturbance

input vector, aφ is the control input, and

.

.

..X

η

η

=

and .Xη

η

=

(3.11)

Two types of sensor output equations have been considered for mechanical

and thermo-mechanical loading. The sensor output equation [150] for

mechanical loading can be written as

33

0[ ]y C X= (3.12)

where output matrix 0[ ]C depends on the modal matrix [ ]ψ and the sensor

coupling matrix us

Kφ . And the sensor output equation for thermo-

mechanical loading has been proposed as

1

[0]s us

y K K dφφ φ

− = − (3.13)

3.3.1 Controllability index for actuator location

The system controllability is a basis in the modern control theory. Wang and

Wang [95] proposed a controllability index for actuator locations, which was

obtained by maximizing the global control force, and this has been

considered in the present study. The modal control force c

f applied to the

system can be written as

[ ] φ=c a

f B (3.14)

It follows from Eq. (3.14) that

[ ] [ ] φ φ=TT T

c c a af f B B (3.15)

Using the singular value analysis, [ ]B can be written as [ ] [ ][ ][ ]=T

B M S N

where [ ] [ ] [ ]=T

M M I , [ ] [ ] [ ]=T

N N I and [ ]

1 0

0

0 0

σ

σ

=

K

O M

M K

K

an

S

where a

n is the number of actuators. Equation (4.15) can be rewritten as

[ ][ ] [ ][ ] φ φ=T TT T

c c a af f N S S N or

2 2 2φ=

c af S

(3.16)

Thus, maximizing this norm independently on the input voltage φa

induces

maximizing2

S . The magnitude of σi is a function of location and the size of

34

piezoelectric actuators. Wang and Wang [1] proposed that the controllability

index is defined by

1

σ=

Ω = ∏an

ii

Maximize (3.17)

The higher the controllability index, the smaller will be the electrical

potential required for control. The control spillover effects are a significant

problem of active vibration control implementation on real structures.

Therefore, a similar controllability index has been proposed in the present

work incorporating residual modes of system/structures as follows

1 1

σ γ σ= =

′Ω = ∏ − ∏a an n

R

i ii i

Maximize (3.18)

where σ R

i are the components of [ ]RS corresponding to residual modes and

γ ′ is a weight

3.4 GA for Optimal Placement

Most natural representation in the form of a string of integers

specifying the locations of actuators has been used in this study. An integer

coded genetic (IGA) algorithm with uniform crossover and mutation have

been developed for optimal placement of actuators. In the present problem

the design variables are the positions of the actuators, and are represented in

a string of integers specifying the locations of actuators. The gene code is

taken as 1 2, ,......., ,......,aj n

ac ac ac ac , where (1, )∈j

ac m and is a positive integer

number and m is the total number of locations for actuators in the

structures/system. Uniform crossover and new mutation techniques for

integer coded genetic algorithm have been discussed in the following

subsections.

35

3.4.1 Uniform crossover

The steps involve in this crossover are

a) A random mask is generated

b) The mask determines which bits are copied from one parent and

which from the other parent

c) Bit density in mask determines how much material is taken from the

other parent

For example, if the randomly generated mask is 0110011000 and parents

are1010001110 and 0011010010 then their offspring will be 0011001010

and 1010010110.

3.4.2 Mutation

A one-digit positive integer value [1, ]j

ac m∈ is generated at random,

which replaces the old one when mutating. If j

ac is equal to old one, then a

new positive integer is selected again until they are different in the

chromosome. The efficiency of the mutation could be improved greatly

using the method.

3.4.3 Optimal Placement using IGA

The fitness value i.e. measure of controllability for the optimal

actuators location has been proposed as follows

36

1

1 1 11

-12

1 1

,

,

10 ,

a

a a a a

a a

n

ii

n n n nR R

i i i ii i ii

n nR

i ii i

without control spillover

ifconsidering control spillover

otherwise

σ

σ γ σ σ γ σ

σ γ σ

=

= = ==

= =

∏

Ω = ′ ′∏ − ∏ ∏ > ∏

′∏ − ∏ ×

(3.19)

The outline of optimization problem using IGA is as follows:

i) Initial chromosomes depending on the number of actuators and

populations are chosen randomly.

ii) The fitness value (measure of controllability) is calculated for each

chromosome.

iii) Genetic operators are applied to produce a new set of chromosomes.

iv) Steps (ii) to (iii) are repeated until the fitness converge

v) The computation is terminated after convergence of fitness and the

chromosome based on the best controllability value is selected as the

optimal locations of actuators.

37

CHAPTER 4

RESULTS AND DISCUSSIONS

Based on the formulations discussed above, a computer code has been

developed for finite element analysis of smart shell structures followed by

optimal actuator placement.

4.1 Structural validation

In order to verify the finite element code developed, a spherical shell

made of graphite/epoxy with the four edges simply supported, having the

following dimensions have been considered: a/b=1, R1=R2=R, R/a=3,

a/h=10. Graphite/epoxy properties considered are as follows: E1=25E2,

G12=G13=0.5E2, υ12=0.25, G23=0.2E2. A 10×10 finite element mesh has been

used to model this entire shell.

4.2 Validation for optimal actuators placement

A smart fiber reinforced polymer (FRP) cantilever beam made of

GR/E has been considered to validate the code for optimal placement of

actuators as well as to compare the performances of integer and binary-

coded GA in terms of generation required to reach the optimal solution. In

this analysis, four actuators and first mode of vibration have been

considered. The length and width of the beam are taken as 0.2 and 0.01 m,

respectively. The stacking sequence of the laminated beam structure

considered is [p/[0/0]s/p]]. Here ‘p’ stands for piezo-patches one for sensing

and the other for actuation. Thickness of each ply has been considered as

0.15 mm and that of piezo-patch is 0.5 mm. The mechanical, electrical and

38

coupled material properties used in the present study have been listed in

Table 1. Several important parameters used for integer- and binary-coded

GA have been listed in Table 2. Optimal actuators placement based on the

maximum controllability index is shown in Fig. 5. It could be clearly

observed from Fig. 5 that the optimal locations of PZT actuators are at the

root of the beam. This result is expected since the curvature of the first mode

of vibration reaches its maximum value at the fixed end of the cantilever

beam and a similar observation was also reported by Wang and Wang .Fig. 6

shows the convergence plot with number of generations for integer-coded

and binary-coded GA and it could be observed that while the integer-coded

GA converges at 31 generations, binary-coded GA converges only after 246

generations.

Table 1: Material properties of structural laminate and PZT

Material properties Structural laminate PZT

E1 172.5 GPa 63.0 GPa

E2=E3 6.9 GPa 63.0 GPa

G12=G13 3.45 GPa 24.6 GPa

G23 1.38 GPa 24.6 GPa

υ12=υ13=υ23 0.25 0.28

ρ 1600 kg m−3

7600 kg m−3

e31=e32 0.0 10.62 C m−2

11= 22= 33 0.0 0.1555×10−7

F m−1

39

Table 2.Several important parameters for integer-and binary-coded GA

Number of genes to represent Integer-coded GA Binary-coded GA

One actuator 1 8

Length of the chromosome 4 32

Population size 10 10

Crossover probability 0.9 0.9

Mutation probability 0.1 0.1

Fig. 5. Optimal location of four actuators on the beam substrate based

on maximum controllability

40

Fig. 6. Comparison of variation of controllability index with generation

for the cantilever beam using integer- and binary-coded GA.

4.3 Optimal vibration control of a semi-circular ring

A simply supported smart FRP composite semi-circular arch under the

action of impulse load at the center has been considered. The radius R1 and

R2 of this panel have been considered to be 0.06 m and infinity respectively.

The dimensions of the base are a=2R1, a, b are the width of the base. The

stacking sequence of the laminated spherical structure considered is

[p/[0/90]s/p]. Here ‘p’ stands for piezo-patches one for sensing and the other

for actuation. Fig. 7 shows optimal actuators placement based on the

maximum controllability index considering six actuators. Fig. 8 presents the

evolution of the best fitness value i.e. controllability index using GA after 50

generations. In this case, the maximum value of controllability index is

0.037 as shown in Fig. 8.

41

Fig. 7. Actuators location on the semicircular ring based on maximum

controllability index.

Fig. 8. Variation of controllability index with generation for

semicircular ring.

42

4.4 Optimal vibration control of laminated spherical shell panel

A simply supported smart FRP composite shell panel on a square base

(a=b=0.02 m) under the action of impulse load at the center has been

considered. The radius (i.e. R1=R2=R) of this panel has been considered to be

0.06 m. The stacking sequence of the laminated spherical structure

considered is [p/[0/90]s/p]. Here ‘p’ stands for piezo-patches one for sensing

and the other for actuation. Thickness of each ply has been considered to be

0.25 mm and that of piezo-patch has been taken as 0.5 mm. A 10×10 finite

element mesh has been considered to model this entire panel. Two types of

piezo-patch locations viz. Placement1 has been considered to study

influence of optimal placement on the input voltage of actuator and the

closed-loop damping ratio. Placement1 stands for optimal actuators

placement based on the maximum controllability index considering six

actuators as shown in Fig. 10. Fig. 11 presents the evolution of the best

fitness value i.e. controllability index using GA after 50 generations. In case

of Placement1, the maximum value of controllability index is 0.680956 as

shown in Fig. 11.

43

Fig. 9 Schematic Representation of a shell panel

Fig.10. Actuators location on the spherical panel based on maximum

controllability index.

44

Fig.11. Variation of controllability index with generation for spherical

panel.

45

CHAPTER 5

CONCLUSION AND SCOPE FOR FUTURE WORK

5.1 Conclusion

In this project we have developed an improved integer-coded GA-

based program for optimal placement of actuators for active vibration

control of smart FRP composite shell structures. This has been used in

conjunction with the developed layered shell finite element procedure for

coupled electromechanical analysis of smart shell structures. The present

integer-coded GA-based optimal actuator location is especially

advantageous for large structures where number of actuators is large. It has

been observed that the proposed improved GA module leads to optimal

locations of actuators. In this project we have used Integer-Coded GA for

finding optimal placement of actuators for a semi-circular ring and a

spherical shell and we have found better results, in the form of higher

controllability as compared to previous work done in this field.

5.2 Scope for Future Works

Multi-objective optimization where both structural design as well as

control performance will be optimized

Non-collocated sensors and actuators optimal locations

Optimal sensors and actuators placement of large structures

requiring large number of sensors and actuators

Parallelizing the optimal placement evaluation

46

CHAPTER 6

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